We successfully obtained the first optical spectra of the faint light echoes around Cassiopeia A and Tycho Brahe's supernova remnants (SNRs) with FOCAS and the Subaru Telescope. We conclude that Cas A and Tycho's SN 1572 belong to the Type IIb and normal Type Ia supernovae, respectively. Light echo spectra are important in order to obtain further insight into the supernova explosion mechanism of Tycho's SN 1572: how the Type Ia explosion actually proceeds, and whether accretion occurs from a companion or by the merging of two white dwarfs. The proximity of the SN 1572 remnant has allowed detailed studies, such as the possible identification of the binary companion, and provides a unique opportunity to test theories of the explosion mechanism and the nature of the progenitor. Future light-echo spectra, obtained in different spatial directions of SN 1572, will enable to construct a three-dimensional spectroscopic view of the explosion.